Mathematical Methods for Physics and Engineering - Matematica.NET

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONSSince p = dy/dx = c 1 , if we substitute (14.41) into (14.39) we find c 1 x + c 2 =c 1 x + F(c 1 ). Therefore the constant c 2 is given by F(c 1 ), and the general solutionto (14.39) isy = c 1 x + F(c 1 ), (14.42)i.e. the general solution to Clairaut’s equation can be obtained by replacing pin the ODE by the arbitrary constant c 1 . Now, considering the second factor in(14.40), we also havedF+ x =0, (14.43)dpwhich has the form G(x, p) = 0. This relation may be used to eliminate p from(14.39) to give a singular solution.◮Solvey = px + p 2 . (14.44)From (14.42) the general solution is y = cx + c 2 . But from (14.43) we also have 2p + x =0 ⇒ p = −x/2. Substituting this into (14.44) we find the singular solution x 2 +4y =0.◭Solution method. Write the equation in the form (14.39), then the general solutionis given by replacing p by some constant c, as shown in (14.42). Using the relationdF/dp+x =0to eliminate p from the original equation yields the singular solution.14.4 Exercises14.1 A radioactive isotope decays in such a way that the number of atoms present ata given time, N(t), obeys the equationdNdt = −λN.If there are initially N 0 atoms present, find N(t) atlatertimes.14.2 Solve the following equations by separation of the variables:(a) y ′ − xy 3 =0;(b) y ′ tan −1 x − y(1 + x 2 ) −1 =0;(c) x 2 y ′ + xy 2 =4y 2 .14.3 Show that the following equations either are exact or can be made exact, andsolve them:(a) y(2x 2 y 2 +1)y ′ + x(y 4 +1)=0;(b) 2xy ′ +3x + y =0;(c) (cos 2 x + y sin 2x)y ′ + y 2 =0.14.4 Find the values of α and β that make( 1dF(x, y) =x 2 +2 + α )dx +(xy β +1)dyyan exact differential. For these values solve F(x, y) =0.484